International Journal of Chemical Engineering

Volume 2017, Article ID 4824376, 10 pages

https://doi.org/10.1155/2017/4824376

## Study on Influence of Fluid Parameters on Axial Coupled Vibration of Pipeline Conveying Multiphase Flow

Department of Petroleum Supply Engineering, Logistical Engineering University, Chongqing 401311, China

Correspondence should be addressed to Ming Chen; moc.621@1028nehcnehc

Received 15 November 2016; Revised 7 April 2017; Accepted 9 May 2017; Published 12 June 2017

Academic Editor: Dmitry Murzin

Copyright © 2017 Ming Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Taking a slurry reservoir-pipeline-valve system as research object, axial dynamic vibrations of pipe system were induced by coupled hydraulic transient due to rapid closure of valve at the end of pipe. The influences of fluid parameters in multiphase flow, including void fraction, density ratio, and elastic modulus ratio between solid phase and liquid phase, on vibration behaviors of pipe system were analyzed. Results of this study show that wave velocities of pressure and stress can be attenuated evidently when void fraction in multiphase fluid is increased appropriately; meanwhile, the amplitudes of pressure fluctuation and pipe vibration are also weakened obviously. With the increase of density ratio between solid phase and liquid phase, the vibrational intension of pipe system becomes stronger and stronger. In this instance, the increments of vibrational energy mainly concentrate in fluid, which leads the pressure energy of fluid to rise up quickly. When elastic modulus ratio between solid phase and liquid phase increases, the total elasticity of fluid decreases gradually. At the same time, both pressure energy of fluid and vibrational intension of pipe are enhanced but the increments are very slight.

#### 1. Introduction

As a kind of the excellent transportation mode, piping systems are widely used in many fields such as marine engineering, nuclear industries, and petroleum engineering. In the operational process of piping system, an extreme hydraulic state, that is, water hammer, is often induced by the disturbance of dynamic or controlling system. Due to the effect of fluid-structure interaction (FSI), water hammer may cause intense coupled vibration of piping system, which decreases the reliability and performance of system and even will lead to serious disasters further. Therefore, it is of great and immediate significance to accurately analyze the coupled vibration principle of piping system for taking safety precautionary measures, ensuring reliable operation of the whole systems, reducing the loss of energy sources, and so on.

Extensive investigations have been carried out on coupled vibration of piping system in the past years. Several coupled vibration models describing nonlinear dynamic behaviors of liquid-conveying pipes were developed by Paidoussis and Li [1], Zhang and Huang [2], Gorman et al. [3], Lee and Chung [4], and Omer et al. [5], respectively. To obtain the coupled vibration response results of piping system in time and frequency domain, different numerical methods, such as MOC, FEM, MOC-FEM, traveling wave method, and transfer matrix method, were used by Wiggert and Tijsseling [6], Kochupillai et al. [7], Zanganeh et al. [8], Ren et al. [9], and Xu et al. [10], respectively.

By virtue of different theoretical models and numerical procedures, some specific researches were concerned with the influence of fluid or structural parameters on the coupled vibration characteristics of piping system. Zhang [11] conducted an investigation into the effects of fluid and shell parameters on the coupled frequencies based on wave propagation approach. Li et al. [12] analyzed the factors affecting the characteristics of FSI by changing the bend radius and angle of curved pipe. Adamkowski et al. [13] carried on their research on the influence of dynamic Poisson effect onto pressure records during hydraulic transient with FSI using experimental data and numerical computations. Yang and Fan [14] studied the influences of pipe structural damping, pipe Poisson’s ratio, and pipe wall thickness on vibration responses of the RPV system. Liu et al. [15] analyzed influences of steam parameters, that is, steam pressure and velocity, on the natural characteristics of steam pipeline systems. Lin et al. [16] discussed the effect of fluid parameters, including liquid pressure, flow velocity, and axial force, on vibration characteristics of hydraulic pipe of aero-engine. Zhang et al. [17] conducted their investigation into the influences of internal flow, the changes of internal flow velocity, and top tension amplitude on coupled vibration of deep-water riser. Eslami et al. [18] studied the effect of aspect ratio of length to diameter on the dynamic response of a fluid-conveying pipe based on the Timoshenko beam model. Their results indicate that the natural frequencies decrease with the increasing of internal fluid velocity and the critical velocity decreases with the decreasing of aspect ratio. Gu et al. [19] studied vibration behavior of a fluid-conveying cracked pipe surrounded by a viscoelastic medium. In their works, the effect of open crack parameters and flow velocity profile shape inside the pipe on natural frequency and critical flow velocity of the system has been analytically investigated. Liu et al. [20] analyzed the influences of bound manner, restraint stiffness, foundation vibration parameters, and structural parameters on pipe outlet pressure fluctuation amplitude. Tian et al. [21] studied vibration characteristics of pipeline under the action of the gas pressure pulsation and the relationships between the gas column natural frequency and orders, the gas pressure pulsation and orders, the exciting force and aspect ratio, and the vibration displacement and velocity of pipeline are acquired. Meng et al. [22] developed a simple FSI model and investigated the effect of internal flow velocity on the cross-flow vortex-induced vibration of a cantilevered pipe discharging fluid.

According to the collected literatures, these researches on coupled vibration of piping system mainly focus on the development and solution of numerical models, the critical flow velocity, parameters resonance of pipe structures, and so on. Furthermore, the pure liquid is often considered as research objects, whereas the liquid containing gas phase and solid phase is rarely taken into account [23]. Actually, with the wide application of mixed transportation technology for multiphase flow or the effect of objective factors (such as interlarding the liquid with some gas or solid impurities), the actual fluid in the pipelines always exists as a kind of multiphase fluids and gas-liquid-solid three-phase mixed flow is the most general and representative in multiphase flows. Thus, this paper takes gas-liquid-solid three-phase mixed flow as research object and the influences of parameters, including void fraction, density ratio, and elastic modulus ratio between solid phase and liquid phase, on vibration characteristics of piping system are studied to illuminate coupled water hammer problems of pipelines further.

#### 2. Mathematical Model

Hydraulic transportation is often used to transport concentrate and tailings in the metallurgical industry as well as cinder in the coal and power industries. To facilitate hydraulic transportation by pipeline, decreasing the wear of the equipment, reducing the conveying velocity, and lowering the operational costs, the fine solid granular materials are usually applied to maintain an even suspended motion in the effect of turbulent diffusion. In such condition, this kind of gas-liquid-solid three-phase mixed flow formed by solid granule, liquid, and some other small amount of gas mixed in is relatively stable, and the concentration of every phase is well-distributed at the cross-section of pipeline. Generally, for such kind of piping system, the flow is treated as homogeneous flow or pseudo-homogenous flow which can be analyzed on the basis of homogeneous flow theory [24].

To obtain a set of convenient FSI governing equations for piping system conveying the mentioned three-phase flow in this paper, the following assumptions are made:(a)Gas phase, liquid phase, and solid phase are mixed evenly, and there are no velocity difference and no mass exchange among the three phases in an adiabatic state.(b)Cross-section changes and deformations along the pipe are small; that is, .(c)The material of pipe wall is isotropic and presents a linear-elastic mechanical behavior.(d)Liquid flashing is not considered in the process of hydraulic transient, and fluid velocity is a cross-section averaged scalar value.

Because many pipes in practice have relatively thick walls, for example, high-pressure pipes in chemical and power industries, the usual assumption of thin-walled pipes is not adopted in this research. Therefore, the ratio of wall thickness to pipe-radius cannot be neglected. Based on the above assumptions, the FSI model can be described by the following expression [25]:where

The characteristic formula for (1) is and its four unequal real roots arewhere .

#### 3. Numerical Scheme

To the numerical simulation of water hammer events without FSI, MOC is always regarded as an excellent numerical strategy. However, for solving FSI model, MOC may introduce excessive numerical dispersion and attenuation in the solutions, which will induce a bad influence on the accuracy of computational results [26]. To avoid the embarrassment, flux vector splitting method is used to solve the proposed FSI model.

Make (1) flux splitting and numerical discretization based on Lax-Wendroff central difference scheme and Warming-Beam upwind difference scheme, both of which possess second-order precision in time and space [27]. According to the literature [28], (1) can be written as the following difference form:where ; ; for or 2, , , , ; for or 4, , , , ; , ; , , , ; is an identity matrix; function is described as follows: when , the value of function equals zero; when , the value of function equals the lesser one of absolute values of and .

The stability condition of the above difference schemes is . These equations are solved subject to boundary conditions at the upstream and downstream ends of the pipeline and initial conditions. For a pipeline connected to a reservoir with constant piezometric head and an unfixed valve at the upstream and downstream ends, respectively, the boundary conditions can be derived from Rankine-Hugoniot condition [29]:

For upstream end,

For downstream end,

#### 4. Discussion about the Influence of Correlative Parameters

A slurry reservoir-pipeline-valve system shown in Figure 1 is taken as a research object in this paper. The pipe and valve are allowed to move freely in the axial direction. The physical parameters of this RPV system are listed in Table 1.